# The Complexity of Computational Problems about Nash Equilibria in Symmetric Win-Lose Games

## TL;DR

This paper proves that deciding the existence of certain Nash equilibria remains NP-hard even in symmetric, win-lose bimatrix games, extending known complexity results to more restricted game classes.

## Contribution

It demonstrates NP-hardness for symmetric win-lose bimatrix games using novel gadgets and symmetrization techniques, showing these restrictions do not simplify the problem.

## Key findings

- NP-hardness holds for symmetric win-lose games
- Developed win-lose gadgets and reduction techniques
- Derived complexity results for search, counting, and parity problems

## Abstract

We revisit the complexity of deciding, given a {\it bimatrix game,} whether it has a {\it Nash equilibrium} with certain natural properties; such decision problems were early known to be ${\mathcal{NP}}$-hard~\cite{GZ89}. We show that ${\mathcal{NP}}$-hardness still holds under two significant restrictions in simultaneity: the game is {\it win-lose} (that is, all {\it utilities} are $0$ or $1$) and {\it symmetric}. To address the former restriction, we design win-lose {\it gadgets} and a win-lose reduction; to accomodate the latter restriction, we employ and analyze the classical {\it ${\mathsf{GHR}}$-symmetrization}~\cite{GHR63} in the win-lose setting. Thus, {\it symmetric win-lose bimatrix games} are as complex as general bimatrix games with respect to such decision problems. As a byproduct of our techniques, we derive hardness results for search, counting and parity problems about Nash equilibria in symmetric win-lose bimatrix games.